Pauli Automorphisms, Clifford Groups, and Local Clifford Groups for Higher-Dimensional Systems

The Clifford and Local Clifford groups for $d > 2$ dimensional systems have been topics of recent interest due to their applications in graph states, quantum codes, and possible applications in fast quantum algorithms. This paper studies these groups more abstractly, by first characterizing the Pauli Automorphism and Local Automorphism groups, and then using these results to determine characteristics of the Clifford and Local Clifford groups. Not only does such an approach reveal new information about the Clifford and Local Clifford groups, but it also shows how many previously derived results arise naturally as simple corollaries. Lastly, we give a systematic method of building an arbitrary Local Clifford operator from a small number of previously known gates, as well as a method to physically implement such an operator.